Globtim.jl Documentation

Global optimization of continuous functions via polynomial approximation

The Problem

Finding all local minima of a continuous function over a bounded domain is fundamentally hard. Standard optimization algorithms (gradient descent, BFGS, etc.) find one local minimum from a given starting point—but how do you know there isn't a better one elsewhere?

The Approach

Globtim solves this by replacing your function with a polynomial approximation. Why polynomials?

1. Smooth functions are well-approximated by polynomials — Chebyshev and Legendre expansions converge rapidly for smooth functions
2. Polynomial critical points can be found exactly — Setting ∇p(x) = 0 gives a polynomial system, which has finitely many solutions that can be computed using homotopy continuation
3. Refinement recovers true minima — Each polynomial critical point seeds a local optimization (BFGS) on the original function

The result: a systematic way to find all local minima, not just the nearest one.

Algorithm Overview

f(x)  →  Polynomial p(x)  →  Solve ∇p = 0  →  Refine with BFGS  →  All minima
         (Chebyshev fit)     (numerical or exact)

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For functions that vary on different scales in different regions, Globtim uses adaptive subdivision to build piecewise polynomial approximations that maintain accuracy everywhere.

Installation

julia> ]
pkg> add Globtim

Additional Dependencies

• Visualization: add CairoMakie or add GLMakie
• Exact solving: Install msolve (symbolic method based on Gröbner basis computations)

Getting Started

To see Globtim in action, run the demo script:

julia --project=. examples/quick_subdivision_demo.jl

This tests adaptive subdivision on multiple functions (sphere, Rosenbrock, Rastrigin) and shows the algorithm's behavior.

For a detailed walkthrough, see Getting Started.